Solution of the Free-Electron Dirac Equation

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

The free-particle Dirac equation provides a nice demonstration of some of the properties of the 4-spinor solutions, but quantum chemistry is mainly concerned with electrons bound in molecules by electromagnetic forces. In a static potential V, such as that provided by the nuclei in the Bom-Oppenheimer frame, where the vector potential A is zero, the time-independent electronic Dirac equation is... 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

The behavior of an electron in an electromagnetic held, in the context of the quantum theory, is determined from the solutions of the Dirac equation. Here the free-particle momentum operator is replaced with the generalized 4-momentum operator, pv + e(Av + Bv). The Dirac equation then takes the form... 

After having derived a truly relativistic quantum mechanical equation for a freely moving electron (i.e., in the absence of external electromagnetic fields), we now derive its solutions. It is noteworthy from a conceptual point of view that the solution of the field-free Dirac equation can in principle be pursued in two ways (i) one could directly obtain the solution from the (full) Dirac equation (5.23) for the electron moving with constant velocity v or (ii) one could aim for the solution for an electron at rest — which is particularly easy to obtain — and then Lorentz transform the solution according to Eq. (5.56) to an inertial frame of reference which moves with constant velocity —v) with respect to the frame of reference that observes the electron at rest. 

In the next section, we will examine some properties of the solutions of the spin-free modified Dirac equation, and we then proceed to a closer inspection of the one-electron operators in the modified formalism before treating the two-electron terms. 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

Since element 103, Lr, is the last member of the actinide or 5f series of elements, element 104 was expected to be the first member of the next group of the periodic table, i.e. the group IV B elements. Indeed, from the results of relativistic Dirac-Fock calculations, the electronic ground-state configuration for a neutral free atom of element 104 was predicted to be 5f 6d 7s [75]. It is expected to have a valence and ionic radius similar to Zr and Hf and to exhibit similar chemical properties [76, 77], Using equations developed by Jorgensen, Penneman and Mann have predicted that element 104 should be predominantly tetravalent in aqueous solution and solid compounds however, the chemistry of element 104 may involve 2+ and 3+ as well [78]. Further, the hydrolytic properties and the solubility of compounds of element 104 are expected to be similar to Hf [75]. Some of the chemical properties predicted for element 104 are given later in Table 13.10. 

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